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ISSN : 2320-2971 (Online)

International e-Journal for Education and Mathematics

www.iejem.org vol. 02, No. 05, (Oct. 2013), pp 29-40

The Sumudu transform and its application to fractional differential

equations

I.A. Salehbhai, M.G. Timol

Department of Mathematics

Veer Narmad South Gujarat University, Surat

Email: [email protected]

Article Info.

A B S T R A C T

Received on 20 June. 2013

Revised on 10 Sept. 2013

Accepted on 21 Sept. 2013

The Sumudu transform provides powerful operation methods for solving

differential and integral equations arising in applied mathematics,

mathematical physics, and engineering science. An attempt is made to solve

some fractional differential equations using the method of Sumudu

transform.

AMS Subject classification: 26A33, 44A10

Keywords:

Integral transform,

Fractional Derivatives,

Sumudu transforms,

Fractional differential

equations.

1. Introduction:

The methods of Integral transforms (Sneddon [8]) have their genesis in nineteenth century

work of Joseph Fourier and Oliver Heaviside. The fundamental idea is to represent a function

f x in terms of a transform ( )F p is

( ) ,

b

a

F p K p x f x dx

(1.1)

where the functions ,K p x is called kernel.

The Sumudu transform and its application to fractional differential equations.

30

There are a number of important integral transforms including Fourier, Laplace, Hankel,

Laguerre, Hermit and Mellin transforms. They are defined by choosing different kernels

,K p x and different values for a and b involved in (1.1).

In 1993, G. K. Watugala [9] introduced the sumudu transform to solve differential equations

and control engineering problems. Weerakoon [10] has discussed this transform by deriving

the Sumudu transform of partial derivatives.

The Sumudu Transform is defined as follows (Watugala [9]):

If f t is a function defined on the Real line, then Sumudu Transform of f t is defined by

0

;Re 0

t

pe

F Sp f t f t dt pp

(1.2)

There has been a great deal of interest in fractional differential equations (Miller and Ross [3],

Oldham and Spanier [4]). These equations arise in mathematical physics and engineering

sciences. There are many definitions of fractional calculus are given by many different

mathematicians and scientists (see Podlubny [5]). Here, we formulate the problem in terms of

the Caputo fractional derivative (see Caputo [1],[2]), which is defined as:

If is a positive number and n is the smallest integer greater than such that 1n n ,

then the fractional derivative of a function f t is defined by (see Podlubny [5]):

1

0

1nt

n

f xd fC f t dx

dt n t x

(1.3)

Further we used the result due to (Katatbeh and Belgacem [6]):

1

0

0r

nr

r

d fS F p f

tp p

d

(1.4)

I.A. Salehbhai and M.G. Timol/Int. e- J. for Edu. & Math. vol. 02 (Oct. 2013) 29-40

31

where n is the smallest integer greater than .

2. Definitions, Notations & Some Results of Special functions

Some special functions used in this thesis are as given below (See Rainville [7]) :

The gamma function is defined as

1

0

x nn e x dx

where, Re 0n (2.1)

Here note that (2.1) can be extended to the rest of the complex plane, excepting zero and

negative integer.

Alternative definitions for Gamma Function (Rainville [7]) are

!lim

1 2 ...

n

z

z zn

n n n n z

(2.2)

1

1

1nn

r

r

e nn e

n r

(2.3)

where is known as Euler’s constant and is defined as following:

1 1 1lim 1 ... log

2 3nn

n

. (2.4)

The Pochhammer symbol n

(Rainville [7]) is defined by the equations

1 2 ... 1 , 1n

n n (2.5)

which is Generalization of factorial function.


32

If is neither zero nor a negative integer, then we can define

n

n

. (2.6)

The ordinary binomial expression (Rainville [7]), defined as

0

( )(1 )

!

na n

n

a zz

n

. (2.7)

The Hypergeometric Function (Rainville [7]) is defined as

2 1

0

, ; ;!

n

n n

n n

a b xF a b c x

c n

(2.8)

where the series on R.H.S. of (2.8), when c is neither zero nor a negative integer, is

absolutely convergent within the circle of convergence 1x , and divergent outside it; on the

circle of convergence the series is absolutely convergent if Re 0c a b .

The Generalized Hypergeometric Function (Rainville [7]) is defined as

11 2p

1 20

1

, ,..., ; , ,..., ; !

p

i nnip

q qq

n

j nj

axa a a

F xb b b nb

. (2.9)

Here note that:

1. If p q , the series (2.9) converges absolutely for every finite x .

2. If 1p q , the series (2.9) converges absolutely when 1x and diverges when

1x .

3. If 1p q , and 1x , the series (2.9) converges absolutely when


33

1 1

Re 0q p

j i

j i

b a

.

4. If 1p q , the series (2.9) diverges for 0z , and for 0z its value is one.

3. The solution of fractional Differential Equations:

In this section, we obtain the solution of some fractional differential Equations using Sumudu

Transform.

(A) Consider the fractional Differential Equations is of the form

5 3 1

2 2

5 3 1

2 2 2

2

coshd d d

t

dt d

x x

d

x

t t

with initial condition 0 0; 0 0; 0 0x x x (3.1)

Solution: Applying the Sumudu Transform (1.2) on both the sides of equation (3.1) and using

(1.4), we have

5 3 1

2 2 2

5 3 1

2

3 1 1

2

2

2

22 2

0 0 0

0 0 01

1

p p p p p x p x p x

p x p x pp

X

x

(3.2)

Further simplification yields,

2

5 3 1

2 2 2

1

1 pp

p p p

X

(3.3)

Taking inverse Sumudu Transform of (3.3) gives


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1

21

2

0

1

21

2

0

1

23

2

0

1

23

2

0

1 1e cos 3 3 sin 3

2 21e

6

1sin 3 e

1 12e e erf

63

1 1e 3cos 3 3 sin 3

2 21e

6

1sin 3 e

1 2e

3

1e

2

u

tt

u

ttt

u

tt

u

tt

x t

t u t u

du

t u

du tu

t u t u

d

u

uu

t u

duu

i

erft i t

(3.4)

Equation (3.4) is the solution of (3.1)

Where erf t is the well-known error function (see Rainville [7]).

(B) Consider the fractional Differential Equations is of the form

3 1

2 2

103 1

2 2

2 logd d

t

dt d

y

t

y with initial condition 0 1; 0 1y y (3.5)


(1.4), we have

3 1

2 2

3 1 1

2 2 20 0 2 0 log2 ep p p p y p ppY y y

(3.6)


3 1

3 1

2

2

2

2 3

2

loge p pp

p p

pY

(3.7)


35


2 2

2

0

2e e2 ln 4 e 2erf 2 8

4

ut t

tty t u du t t i i tu

………...(3.8)

Where t is the well-known Dirac Delta function (see Rainville [7]).

Equation (3.8) is the solution of (3.5)

(C) Consider the fractional Differential Equations is of the form

3 1 1

2

7

2 2

2 2

5 3

2

2 td d de

dt dt

z

dt

z z with initial condition 0 1; 0 1; 0 1; 0 1z z z z (3.9)


(1.4), we have

7 5 3 7 5 3 1

2 2 2 2 2 2 2

5 3 1 3 1

2 2 2 2 2

0 0 0 0

12 0 2 0 2 0 0 0

1

2p p p Z p p z p pz z z

z

p

p p p p pz z z zp

(3.10)


7 5 3 1

2 2 2 2

7 5 3

2 2 2

13 4 4

2

1 pZ

p p p p

p

p p p

(3.11)



36

5

2

1 1

1 1erf e 4 erf 8 e

4 4

4 72, ;

215

t ttz t t i i t t

tF t t t

(3.12)

Hence, (3.12) is the required solution of (3.9).

(D) Consider the fractional Differential Equations is of the form

7 5 3

2 2

3

2 2 2

2

1 12 3 sin

d d dt

dt dt dt

z z z with initial condition 0 0 0 0 0z z z z (3.13)

Solution: Applying the Sumudu Transform (1.2) on both the sides of equation (3.13) and

using (1.4), we have

7 5 3 7 5 3 1

2 2 2 2 2 2 2

5 3 1 3 1

2 2 2 2 22

2 3 2 0 2 0 2 0 2 0

3 0 3 01

3 0 0 0

z zp p p A p p z p p p

p

z

z z z zp zp p p pp

(3.14)


2

7 5 3

2 2 2

1

2 3

p

pp

p p p

A

(3.15)


32 25 2

1

/ 2

2

1 2 2

9 5 7150 330 8 1; , ; 60 1; , ;

4 4 4

275 e erf 240 2e erf

2

7

4 4 4

t

t

Ft t

a Ft t t t

i ti i t i

(3.16)

Hence, (3.16) is the required solution of (3.13).


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4. Some Graphs:

Figure 4.1: Plot of

z t

Figure 4.2: Plot of

Re z t


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Figure 4.3: Plot of

Im z t

Figure 4.4: Plot of

Re a t


39

Acknowledgement

First author is thankful to University Grants Commission, New Delhi for awarding Dr. D. S.

Kothari Postdoctoral Fellowship (Award No.: File no. F.4-2/2006 (BSR)/13-803/2012 (BSR)

dated 09/11/2012).

References

[1] Caputo M. , Linear models of dissipation whose Q is almost frequency independent-

part II, Geophysical Journal of the Royal Astronomical Society, vol. 13, pp. 529–539,

1967.

[2] Caputo M. , Elasticity and Dissipation, Zanichelli, Bologna, Italy, 1969.

[3] Miller K.S., Ross B., An Introduction to the Fractional Calculus and Fractional

Differential Equations, John Wiley and Sons, Inc., 1993.

[4] Oldham K.B., Spanier J., Fractional Calculus: Theory and Applications,

Differentiation and Integration to Arbitrary Order, Academic Press, Inc., New York-

London, 1974.

[5] Podlubny I., Fractional Derivatives: History, Theory, Applications. Utah State

University, Logan, September 20, 2005.

[6] Katatbeh Q. D. and Belgacem F.B.M., Applications of the Sumudu transform to

fractional differential equations, Nonlinear Studies 18 (2011) 99-112.

[7] Rainville E.D., Special Functions, The Macmillan Company, New York, 1960.

[8] Sneddon I.N., The Use of Integral Transforms, McGraw-Hill Book Company, New

York, 1972.


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[9] Watugala G. K., Sumudu transform: a new integral transform to solve differential

equations and control engineering problems, International Journal of Mathematical

Education in Science and Technology 24 (1993) 35–43.

[10] Weerakoon S., Application of Sumudu transform to partial differential equations,

International Journal of Mathematical Education in Science and Technology 25 (1994)

277–283.

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